Question: Determine how many solutions exist for the system of equations. ${-3x-y = -1}$ ${6x+2y = 2}$
Answer: Convert both equations to slope-intercept form: ${-3x-y = -1}$ $-3x{+3x} - y = -1{+3x}$ $-y = -1+3x$ $y = 1-3x$ ${y = -3x+1}$ ${6x+2y = 2}$ $6x{-6x} + 2y = 2{-6x}$ $2y = 2-6x$ $y = 1-3x$ ${y = -3x+1}$ Just by looking at both equations in slope-intercept form, what can you determine? ${y = -3x+1}$ ${y = -3x+1}$ Both equations have the same slope and the same y-intercept, which means the lines would completely overlap. ${1}$ ${2}$ ${3}$ ${4}$ ${5}$ ${6}$ ${7}$ ${8}$ ${9}$ ${\llap{-}2}$ ${\llap{-}3}$ ${\llap{-}4}$ ${\llap{-}5}$ ${\llap{-}6}$ ${\llap{-}7}$ ${\llap{-}8}$ ${\llap{-}9}$ ${1}$ ${2}$ ${3}$ ${4}$ ${5}$ ${6}$ ${7}$ ${8}$ ${9}$ ${\llap{-}2}$ ${\llap{-}3}$ ${\llap{-}4}$ ${\llap{-}5}$ ${\llap{-}6}$ ${\llap{-}7}$ ${\llap{-}8}$ ${\llap{-}9}$ Since any solution of ${-3x-y = -1}$ is also a solution of ${6x+2y = 2}$, there are infinitely many solutions.